Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 262–274
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Colloids and Surfaces A: Physicochemical and
Engineering Aspects
journal homepage: www.elsevier.com/locate/colsurfa
Modeling dynamic interaction between an emerging water droplet
and the sidewall of a trapezoidal channel
Preethi Gopalan a , Satish G. Kandlikar b,a,∗
a
b
Department of Microsystems Engineering, Rochester Institute of Technology, Rochester, NY 14623, USA
Department of Mechanical Engineering, Rochester Institute of Technology, Rochester, NY 14623, USA
h i g h l i g h t s
g r a p h i c a l
a b s t r a c t
• Geometric analysis performed to
•
•
•
•
predict the droplet behavior near
channel corner.
Droplet dynamics depends on corner
angle, contact angle and droplet inlet
location.
Model predicts droplet jumping
behavior as a function of these
variables accurately.
Model was verified with experiments
using two base and four sidewall
materials.
Surface energy analysis also was
performed to understand droplet
jumping behavior.
a r t i c l e
i n f o
Article history:
Received 13 July 2013
Received in revised form 30 August 2013
Accepted 4 September 2013
Available online 16 September 2013
Keywords:
Droplet jumping
Corner filling
Gas channel
PEMFC
Contact angle
Surface energy
a b s t r a c t
Water droplet–sidewall interactions at the corner of the two walls of different surface energies are a
significant problem from a fundamental perspective. Such conditions are encountered in many diverse
applications. In proton exchange membrane fuel cells (PEMFCs), removal of water droplets emerging
from the base of a horizontal gas diffusion layer (GDL) into the gas channel has been a major issue
from water management perspective. Recent studies have provided valuable understanding on the water
droplet behavior near the GDL/sidewall corner. Two distinct behaviors of filling and non-filling of the gas
channel corners were observed by the previous investigators when the emerging droplet interacted with
the sidewall. However, authors’ group also observed a new condition, droplet jumping to the sidewall
under specific combinations of the corner angle, and the base and sidewall contact angles. The focus of
the present work is to gain a fundamental understanding of the droplet–sidewall interactions in terms of
the droplet jumping, corner filling, and non-filling behaviors without airflow. This work is complemented
with an experimental study by varying the location of the droplet emergence, the corner angle, and the
surface energies of the walls.
© 2013 Elsevier B.V. All rights reserved.
1. Introduction
1.1. Water removal from PEMFC gas channels
∗ Corresponding author at: Department of Mechanical Engineering, Rochester
Institute of Technology, Rochester, NY 14623, USA. Tel.: +1 585 475 6728.
E-mail addresses: [email protected] (P. Gopalan), [email protected]
(S.G. Kandlikar).
0927-7757/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.colsurfa.2013.09.013
Over the past few decades, the water management aspect of
PEMFCs has been of major interest in the automotive fuel cell
application [1–14]. Despite the many advantages of PEMFCs, one
major issue preventing their widespread commercialization in the
P. Gopalan, S.G. Kandlikar / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 262–274
automotive sector is the water management at the startup and the
shutdown stages [1]. During these stages, the gas flow rates through
the channels are relatively low, which makes it difficult to remove
the liquid water present in the channels. Therefore, effective water
removal at these conditions is very important for satisfactory operation and higher fuel cell operating efficiencies [2].
It is revealed that water accumulation at the corners where
the GDL and the channel walls of the gas channel meet leads to
increased pressure drop in the channels [15,16]. Water stagnation
in the gas channels hinders the reactant gas flow to the catalyst
layer where the reaction occurs, and lowers the overall efficiency
of the cell [3,4]. Water removal from the gas channel is dependent
on several factors such as gas velocity, contact angle on the GDL and
channel wall surfaces, contact angle hysteresis, droplet emergence
location, and the cross sectional geometry of the channel [1–23].
Gas flow in the channel exerts the pressure force on the droplet
modifying its shape, detaching it from the GDL and the channel wall
surfaces, and removing it from the gas channel [9–12]. Some of the
works done previously to incorporate the effect of aforementioned
parameters are discussed further in the remainder of this section.
1.2. Forces affecting droplet removal
Several researchers have studied the influence of different forces
on the droplet removal from the base within a microchannel.
Chen et al. provided simplified models for predicting the water
droplet removal from the GDL based on a macroscopic force balance [9]. They found that the droplet removal could be enhanced
by increasing the flow channel length or mean gas flow velocity,
in combination with decreasing the channel height or the contact
angle hysteresis, as well as increasing the hydrophobicity of the
GDL. Golpaygan and Ashgriz conducted a numerical study to understand the effect of liquid droplet and the gas flow properties on the
droplet mobility in the gas channel. It was observed that surface
tension has the greatest influence among the parameters affecting
the droplet mobility [10]. Higher the surface tension, more difficult
it is to remove the droplet from the microchannel. Theodorakakos
et al. investigated numerically the detachment of a liquid droplet
from the GDL surface under the influence of air in cross-flow [11].
They provided a correlation to find the critical air velocity, for a
given droplet size, at which the droplet is removed from the GDL.
This data was compared with the experimental results, and the
adhesion force between the liquid droplet and the solid surface was
estimated. To get a clear understanding of the effects of different
forces on droplet removal, Cho et al. conducted a 3D numerical analysis [6,7]. They found that the viscous force has a significant impact
on smaller droplets at lower air velocities, whereas the pressure
drag is dominant for larger droplets. From all these works, the GDL
and the channel wall surfaces are seen to have a major influence
on the liquid water removal from the gas channels.
1.3. Effect of GDL surface energy on droplet removal
It has been reported previously that the surface energy of the
GDL affects the droplet removal condition within the PEMFCs gas
channels [12,24–26]. He et al. developed a two-fluid model, and
presented another simplified numerical model to obtain the droplet
detachment diameter that takes into account the effect of surface
properties of the GDL [24]. The results showed that a lower surface
tension and a higher contact angle of the liquid at the GDL-channel
interface are beneficial for water removal as they result in smaller
droplet sizes at the detachment. Tuber et al. used a transparent
material for the gas channel walls to elucidate the effect of GDL
materials on the water management in PEMFCs [26]. They reported
that hydrophilic GDLs spread the water on the GDL surface and lead
to higher water accumulation in the gas channels. This increases the
263
pressure drop in the channel and decreases the fuel cell efficiency.
Kumbur et al. developed a theoretical model to predict the influence
of GDL hydrophobicity and contact angle hysteresis on the mobility
of the droplet and its removal [12]. It was seen that at high gas flow
rates, the surface hydrophobicity of the GDL helps in the removal
of the droplet. At low gas flow rates, hydrophobicity of the GDL has
only a small influence on droplet removal.
1.4. Effect of channel material properties on droplet removal
A few studies are reported in literature to address the effect
of channel material properties on the water accumulation over
the GDL surface. It was noted that hydrophilic channel surfaces
facilitate the water droplet removal by wicking the water into the
channel corners [3]. Lu et al. performed ex situ experiments to
investigate the effect of different channel wall materials on droplet
removal [22]. They found that the hydrophilic channels promote
uniform distribution of water on the GDL surface. It was also noted
that the hydrophilic channels help in creating film flow, which aids
in lowering the pressure drop compared to the hydrophobic channels. Zhu et al. conducted numerical studies and proposed that the
hydrophilic channel surface leads to film flow in the channel, which
eventually blocks the channel pathway. They also found that the
channel geometry plays an important role in water droplet removal
from gas channel [27,28].
1.5. Effect of channel cross sectional geometry on droplet removal
To understand the importance of channel geometry on water
removal, Zhu et al. conducted simulations with different gas
channel configurations. They found that the droplet detachment
diameter was smaller and the droplet removal time was lower for
triangular and trapezoidal channels as compared to those in the
rectangular and upside-down trapezoidal channels [27,28]. They
also found that the height of the channel as well as the diameter
of the water inlet pore have significant effect on the deformation
and detachment of the droplets. To further elucidate the effect
of channel geometry, Lu et al. performed an experimental study
and concluded that sinusoidal and trapezoidal channel geometries
result in an easier water removal and a lower pressure drop in
the system compared to the rectangular channels [22]. Wang et al.
performed two-phase flow modeling with different channel geometry and reported that water generally gets trapped around the
geometrical heterogeneity [29].
To investigate the water accumulation and removal processes
from the gas channel at a microscopic level, Rath and Kandlikar
analyzed the droplet interactions with the sidewall of a trapezoidal
gas channel at different channel open angles [23]. They found that
the droplet interacted with the channel walls in two ways: droplet
exhibited either fill or not-fill conditions at the corner. They used
Concus–Finn condition to predict the corner filling behavior by
the droplet [30–32]. Rath and Kandlikar also observed a droplet
jumping behavior (droplet would detach from the horizontal base
surface and move completely toward the sidewall of the channel)
at the transition angle between corner filling and non-filling for a
given material pair [33]. Transition angle is a channel open angle
for a given material pair below which the droplet remained pinned
to the sidewall and not-fill the channel corners. However, at angles
greater than the transition angle, the droplet would fill the channel
corners. Gopalan and Kandlikar extended the work on corner filling
condition by imposing an airflow to simulate conditions in a PEMFC
gas channel [15,18]. They noted that at any given air flow rate,
the corner filling in the channel depends upon the instantaneous
dynamic contact angles (IDCA) the droplet makes with the sidewall and the GDL during its oscillatory growth process. Since this
work was confined to a single channel wall and GDL material pair,
264
P. Gopalan, S.G. Kandlikar / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 262–274
Fig. 1. Image sequence of the droplet jumping to the sidewall in a 50◦ channel made of polycarbonate sidewall and SGL-25BC GDL.
they extended this work to investigate the effect of different channel wall and GDL materials on corner filling behavior [16,19]. They
showed that the corner filling of the channel was more dynamic
in nature and the IDCA values for the GDL and the channel wall
need to be incorporated into the Concus–Finn condition to predict
the corner filling behavior in the presence of an air flow. Das et al.
also worked on a similar problem where they analyzed on why the
droplet always remains at a finite distance from the corner and
they referred it to as “enclosure effect” [34]. They mentioned that
the enclosure effect is dependent on the size of the droplet, surface
wettability and the dynamics of the drop evaporation. The authors
also noted that the evaporation flux plays an important role at the
three-phase contact line of the droplet and dictates the enclosure
effect.
Although there have been comprehensive studies reported on
the droplet dynamics under different conditions, most of the theoretical analyses performed to understand the droplet removal
have been done for parallel plates without considering the channel
sidewall interactions. A theoretical analysis of the droplet interaction with the channel walls in a trapezoid channel would be
highly desirable in characterizing the droplet-wall interactions. In
addition, it is noted that the corner filling and non-filling study
is relevant not only in the PEMFC application, but also in other
microfluidic areas. Some of the areas where this work could be
applied are in making optical current transformers [35], athermal arrayed waveguide grating (AWG) [36], biomedical application
(polymer microreplication) [37], and understanding the bubble
growth and rewetting phenomenon during the boiling process [38].
This work is also believed to be useful in designing the micromechanical mercury-contact relay devices [39], plasmon-polariton
propogation devices [40], heat pipes [41] and polymeric hollow
micro-needles [42].
The present work aims at examining the effect of the following parameters on the droplet-wall interactions in a corner of a
trapezoidal channel: droplet diameter, droplet emergence location,
corner angle, and the contact angles of the GDL and channel walls
on the droplet jumping, and corner filling and non-filling behavior.
Since the evaporation time scale for water droplets is significantly
longer than the droplet growth times, its effect is negligible in the
present study. A Gibbs free energy analysis is also performed to
understand the effects of the surface energy of the base and the wall
material on the droplet jumping behavior. The analytical model is
validated with experimental data.
the transition corner angle of 50◦ [33]. They observed that the
droplet did not show either corner filling or non-filling behavior, but instead the droplet detached itself from the base wall and
jumped to the channel sidewall upon contact. Fig. 1 shows the
image sequence of the droplet jumping behavior to the sidewall
in a 50◦ channel made by polycarbonate sidewall and SGL-25BC
GDL. Fig. 1(a) shows the emergence of the droplet from the base
of the channel. Fig. 1(b) shows the droplet as it contacts the sidewall right before it jumps. Once the sidewall contact is made, the
droplet quickly moves toward the sidewall as shown in Fig. 1(c)
and then jumps as shown in Fig. 1(d). Finally, the droplet completely disconnects from the base and hangs from the sidewall as
shown in Fig. 1(e). To characterize the droplet jumping behavior
and the effect of different parameters on droplet jumping, a geometric analysis is performed.
For this analysis, three conditions are examined separately. The
first condition is when the droplet sat on the base and touched the
sidewall right before it jumped as shown in Fig. 2. At this state, the
maximum radius RC1 of the droplet right before it jumped to the
sidewall is determined.
When the droplet jumped to the sidewall, it was completely
detached from the base and hung from the sidewall. Second condition in this analysis is to determine the droplet radius when it was
hanging from the sidewall. Since the droplet volumes just before it
touched the wall and right after it jumped to the wall are the same,
these two volumes are equated to find the radius of the droplet (R2 )
as it hangs from the sidewall.
When the droplet hangs from the sidewall, the maximum size
of the droplet can be large enough to touch the base of the channel
and bridge between the sidewall and the base. Therefore, the third
condition is based on the maximum radius RC2 that the droplet can
2. Analytical model for droplet sidewall interaction
The experimental study performed by Rath and Kandlikar
showed how the droplet behaved near the channel corners for
a horizontal SGL-25BC GDL and a polycarbonate channel wall at
Fig. 2. Description of the different contact lengths and contact angles the droplet
makes with the base just before it jumps to the sidewall.
P. Gopalan, S.G. Kandlikar / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 262–274
265
Fig. 3. (a) Image of the droplet right before it jumps to the sidewall and (b) image of the droplet hanging from the sidewall after it jumped. Point D acts as the mid-point
from where the droplet hangs after it jumps to the sidewall.
have when it jumps to the sidewall and hangs from it, right before
it touches the base and bridges between the two walls.
The droplet jumping behavior to the sidewall can be predicted
by comparing R2 with RC2 . For the droplet to remain hanging from
the sidewall, R2 needs to be smaller than RC2 . Using this analysis,
a criterion is obtained in terms of the droplet volume, the droplet
emergence location, contact angle of the base and the sidewall, and
the trapezoid channel angle.
In the above analysis, the gravitational force is neglected. This
assumption is valid for droplets with the Bond number (ratio of
gravity to surface tension forces) less than 0.3 [13]. For water
droplets at standard earth gravity, this results in a limiting droplet
radius of 1.5 mm. In addition, the droplet jumping to the sidewall
occurred because the base material used had higher a contact angle
compared to the sidewall material used. Therefore, for the droplet
to jump, the surface energy of the sidewall needs to be lower than
the base. To understand the effect of surface energies on the droplet
behavior, a Gibbs free energy analysis is performed on the droplet
under the two states: (a) just before the jump and (b) as it hangs
from the sidewall.
2.1. Condition 1: droplet sitting on the base wall before jumping
to the sidewall
The relationship between the above-mentioned angles is given
in detail in Appendix I.
The channel open angle is related to the droplet critical radius
as
tan 2˛ =
RC1 sin ˇ + RC1 sin ı
(1)
a − RC1 cos ı
From Eq. (1), the maximum radius of the droplet RC1 just before
it snaps and jumps to the sidewall is obtained. Detailed derivation
of RC1 is given in Appendix II.
RC1 =
a sin 2˛
1 − (cos cos 2˛)
(2)
Fig. 3 shows two images from the experimental video where it
is seen that the point D acts as the mid-point around which the
droplet hangs when it is pulled toward the sidewall. The droplet
jumping condition is dependent on the distance x. Detailed derivation of x is given in Appendix II.
2
x = RC2 (cos 2˛ − cos ) + (a − RC1 sin 2˛)2
1/2
1
(3)
Next step is to find the actual radius R2 of the droplet when the
droplet hangs from the sidewall after it jumps.
2.2. Condition 2: droplet hanging from the sidewall after the jump
The maximum radius RC1 that is associated with the droplet volume just before it jumps to the sidewall is first determined. For
this analysis, the following parameters as shown in the Fig. 2 are
defined:
is the static contact angle of the base wall.
2˛ is the open angle between the wall and the base.
a is the distance of the droplet emergence location from the channel corner.
x is the distance from the channel corner to where the droplet
contacts the sidewall (D) right before it jumps.
b is the vertical distance from the center of the sphere which constitutes the droplet to the base of the channel.
c is the distance from center of the droplet to the base such that
it makes an angle ı with the base and is in straight line with the
radius of the droplet.
d and e are the vertical and horizontal distances from the tangent
point D to the center of the droplet respectively.
ˇ is the inner open angle made by the radius RC1 with the base of
the channel.
To find the radius R2 when the droplet hangs from the sidewall,
the volume of the droplet right before it jumped (V1 ), and the volume of the droplet just after it jumped (V2 ) are equated. Volume
of the droplet while sitting on the base right before the jump as
shown in Fig. 2 is given as:
V1 =
(2RC3 − 3RC3 cos + RC3 cos3 )
1
1
1
3
(4)
Volume of the droplet while hanging from the sidewall is given
as
V2 =
3
R (2 − 3 cos + cos3 )
3 2
(5)
where ϕ is the static contact angle of the sidewall. Detailed derivations for V1 and V2 are given in Appendix II.
Equating both the volumes V1 and V2 , R2 is found to be
R2 =
a sin 2˛
1 − (cos cos 2˛)
2 − 3 cos + cos3 1/3
2 − 3 cos + cos3 (6)
266
P. Gopalan, S.G. Kandlikar / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 262–274
Table 1
Transition angles calculated using Concus–Finn condition for different material pair
used for the base and sidewall of the channel.
Fig. 4. Representation of a droplet having the maximum radius (RC2 ) while hanging
from the sidewall such that the droplet just touches the base of the channel. It is
assumed that the droplet is a part of a sphere and also neglecting the gravitational
effect on the droplet since the droplet sizes used are very small.
2.3. Droplet jumping criterion analysis from geometrical
considerations
Next step is to determine the maximum radius RC2 as shown in
Fig. 4. The following parameters shown in Fig. 4 are defined as:
O is the center of the hanging droplet.
x is the distance from the channel corner to the mid-point D around
which the droplet hangs from the sidewall.
g is the vertical distance from the center of the droplet with radius
RC2 to the channel sidewall.
is the angle made by line g with the sidewall at point C.
f is the distance between the points D and C.
z is the distance from the center of the droplet to the mid-point D
on the sidewall.
sin 2˛ =
RC2 − RC2 cos / cos 2˛
(7)
x − RC2 cos tan 2˛
Derivation for Eq. (7) is given in Appendix II.
Therefore by rearranging Eq. (7), RC2 in terms of 2˛ and ϕ is
found.
2
RC2 =
[RC2 (cos 2˛ − cos ) + (a − RC1 sin 2˛)2 ]
1
1/2
sin 2˛
(8)
(1 − cos cos 2˛)
Base material
Sidewall material
Transition angle (◦ )
SGL-25BC
SGL-25BC
PTFE
PTFE
SGL-25BC
SGL-25BC
Polycarbonate
PTFE (Polytetrafluoroethylene)
Polycarbonate
Aluminum
Aluminum
Rough polycarbonate
54
82
20
29
63
51
Eq. (10) gives the condition to determine whether the droplet
would jump to the sidewall or not. For any given material at a
given channel open angle and droplet emergence location, the
droplet behavior in the channel can be identified using this criterion. Concus–Finn condition is able to predict the channel corner
filling and non-filling behavior. However, it is not able to predict the
droplet jumping behavior. Therefore, to understand how a droplet
would behave in a system for a given material pair, firstly check if
R2 /RC2 ≤ 1. If this criterion is satisfied, then the droplet would jump
to the sidewall. If this condition is not satisfied, then the droplet
would lead to corner filling or non-filling condition, which could
be predicted using Concus–Finn condition.
3. Experimental results
To validate the criterion derived for determining the droplet
jumping behavior, experiments were performed on different base
and wall materials. A preferential pore of 100 ␮m was made on
the base material for the droplet to emerge at a known location.
The distance between the preferential pore and the channel corner was varied between 0.5–2.5 mm during the experiments and
the droplet dynamics were observed. The channel open angle was
varied from 25◦ –90◦ . In these experiments, only one side of the
channel wall was used with the GDL base for easier visualization.
The experiments were performed such that the droplet size does
not exceed 1.5 mm in diameter while sitting on the GDL without
touching the walls. The videos were captured using high-speed
Keyence VW-6000 camera at 125 fps and the droplet dynamics
were evaluated. Different set of material pairs used for the base and
channel wall material pairs and their theoretical transition angles
based on the Concus–Finn condition are given in Table 1. The experimental results for the droplet behavior near the channel corners
for different material pairs and different open angles are summarized in Table 2. Fig. 5 shows the plot of R2 /RC2 versus the channel
open angle for different tests performed. Fig. 5 validates the use
Substituting RC1 in Eq. (8)
2
2
2 1/2
(a sin 2˛/1 − (cos cos 2˛)) (cos 2˛ − cos ) + a − (a sin2 2˛/1 − (cos cos 2˛))
RC2 =
(9)
(1 − cos cos 2˛)
It is seen from Eq. (8) that RC2 is a function of RC1 , distance of
the droplet emergence location from the channel corner a, channel
open angle 2˛, and the base and side wall contact angles and
ϕ, respectively. The droplet jumping condition is thus defined as
follows:
(a) R2 ≤ RC2 – the droplet would jump to the sidewall.
(b) R2 > RC2 – the droplet bridges the base and channel wall.
Equating Eqs. (6) and (9), the criterion for droplet jumping is
obtained
R2
=
RC2
sin 2˛
of the criterion with the experimental data for different sets of
materials used. In the plot, points show experimental data and a
dotted line shows the theoretical prediction. The droplet-jumping
criteria for different material pairs were calculated using Eq. (10).
Fig. 5(a) is a plot between the experimental values and the
theoretical predictions for an SGL-25BC base and a polycarbonate
sidewall. All three behaviors, droplet jumping, corner filling and
non-filling were observed during experiments for different channel open angles used. It was observed that the theoretical value
of R2 /RC2 for 40◦ , 50◦ , 52◦ and 60◦ channel corner angles also fell
(a sin 2˛/1 − (cos cos 2˛))[2 − 3 cos + cos3 /2 − 3 cos + cos3 ]
2
2
2
1/3
(1 − cos cos 2˛)
(a sin 2˛/1 − (cos cos 2˛)) (cos 2˛ − cos ) + (a − (a sin 2˛/1 − (cos cos 2˛)))
2 1/2
sin 2˛
≤ 1(10)
P. Gopalan, S.G. Kandlikar / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 262–274
267
Table 2
Comparison of the experimental results for corner filling and non-filling with the Concus–Finn condition.
Material pair
Droplet emergence
location (mm)
Channel angle
Concus–Finn based
transition angle (◦ )
Experimental
results
SGL-25BC base and polycarbonate
sidewall
2.1
2.5
2
3.45
5.6
6.5
1.4
1.6
1.7
3
4.2
4
1.6
1.4
1.7
2.5
4
5.2
2.4
1
1.5
2.9
3.5
1.5
1.7
2
2.9
3.4
1.7
3
2
5.4
4.8
1.5
2.4
4.3
6.3
3.4
2.3
3.1
3.7
5.2
4.56
2.5
3.4
2
4.2
5
1.6
2
2.5
2.9
3.3
4
1.1
2.6
2.5
2.3
3.5
1.7
2.1
2.6
3.1
2.4
1.4
2.6
2.5
4
1.89
0.7
2.3
1.3
2
3.4
0.5
25
30
30
30
30
30
40
45
45
45
45
45
50
52
50
50
50
50
60
60
60
60
60
90
90
90
90
90
30
30
30
30
30
45
45
45
45
45
50
50
50
50
50
60
60
60
60
60
90
90
90
90
90
54
No fill
No fill
No fill
No fill
No fill
No fill
Jump
No fill
No fill
No fill
No fill
No fill
Jump
Jump
No fill
No fill
No fill
No fill
Jump
Jump
Jump
Fill
Fill
Fill
Fill
Fill
Fill
Fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
Fill
Fill
Fill
Fill
Fill
30
30
30
30
30
45
45
45
45
45
50
50
50
50
50
60
60
60
60
60
90
20
SGL-25BC base and PTFE sidewall
PTFE base and polycarbonate
sidewall
82
Fill
Fill
Jump
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
268
P. Gopalan, S.G. Kandlikar / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 262–274
Table 2 (Continued)
Material pair
0.9
PTFE base and aluminum sidewall
SGL-25BC base and aluminum
sidewall
SGL-25BC base and rough
polycarbonate sidewall
Droplet emergence
location (mm)
90
1.5
1.9
0.6
2.5
5
4.8
4
2.8
3.4
5
2.7
3.3
2.5
3.5
2.5
2.4
1.5
2
2.4
2.6
2
1.5
3
1.5
2.7
3
3
3.5
3.3
4.6
5.6
2.5
1
3.1
2.5
3.5
5.6
2.4
1.7
2.4
3.4
3.6
2
2
1.5
3
3.5
1
0.5
1
1.3
2
2.8
0.2
1
1.9
3
3.5
1.7
1
3
4.3
2
1.35
2.2
2.9
3
3.5
1.3
0.4
1
2
2.5
1.5
2
Channel angle
Concus–Finn based
transition angle (◦ )
Experimental
results
Fill
90
90
90
30
30
30
30
30
45
45
45
45
45
50
50
50
50
50
60
60
60
60
60
90
90
90
90
90
30
30
30
30
30
45
45
45
45
45
50
50
50
50
50
60
60
60
60
60
90
90
90
90
90
30
30
30
30
30
45
45
45
45
45
50
50
50
50
50
60
60
60
60
60
90
90
29
63
51
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
Fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
Jump
No fill
No fill
No fill
Jump
No fill
No fill
No fill
No fill
No fill
Jump
Jump
Fill
Fill
Fill
Fill
Fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
No fill
Jump
Jump
Jump
Fill
Fill
Fill
Fill
Fill
Fill
Fill
P. Gopalan, S.G. Kandlikar / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 262–274
269
Table 2 (Continued)
Material pair
2.9
Droplet emergence
location (mm)
90
3.5
1
Channel angle
Concus–Finn based
transition angle (◦ )
Experimental
results
Fill
90
90
below 1 showing that the droplet would jump to the sidewall.
The proposed model is thus seen to predict the droplet jumping
behavior accurately.
Similar experiments were performed with SGL-25BC GDL base
and a PTFE (polytetrafluoroethylene) sidewall and the results are
shown in Fig. 5(b). Here only corner filling and non-filling behaviors were observed during the experiments. For an open angle
below 60◦ , no corner filling of the channel corners was observed in
Fill
Fill
the experiments. At 90◦ open angle, the droplet filled the channel
corners. It is seen from Table 2 that for this material pair, the transition angle from non-filling to filling is at 82◦ . This means that the
droplet would fill the channel corners for open angles above 82◦
and thus matches with the experimental results.
Fig. 5(c) is the plot for a PTFE base and a polycarbonate sidewall. Since the transition angle for this material pair is 20◦ , the
entire range of open angle used for experiments should show
Fig. 5. Graphical representation of the proposed criterion to predict the droplet jumping behavior in a PEMFC gas channel. The experimental points falling below the proposed
criterion shown by dash line leads to droplet jumping to the sidewall. Droplet transition from non-filling to filling of the channel corner based on Concus–Finn condition is
plotted by the dot-dash line. Any points falling on the right side of the dot-dash line shows corner filling of the channel and points falling on the left side of the dot-dash line
shows no corner filling behavior by the droplet. 96% of the theoretically predicted behavior matched the experimental data.
270
P. Gopalan, S.G. Kandlikar / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 262–274
Fig. 6. Image sequence for SGL-25BC base and Aluminum sidewall showing droplet jumping behavior in a 45◦ channel. The droplet radius while hanging from the sidewall
is equal to the maximum radius that droplet could have which leads to bridging between the base and sidewall.
corner filling. It was seen that corner filling behavior was observed
for different open angles used experimentally. However, at 30◦
open angle, the droplet jumped to the channel sidewall. Using the
proposed criterion, the R2 /RC2 value for this case was calculated
and it was observed that the value was below one showing that the
droplet should jump.
Similarly, for a PTFE base and aluminum sidewall, no jumping
behavior was observed for any channel angles used. Only corner filling behavior was observed experimentally for open angles greater
than 30◦ and is shown in Fig. 5(d). According to Concus–Finn condition, channel corner filling should occur for open angles greater
than 29◦ . Thus, experimental results match with the theoretical
predictions.
Fig. 5(e) shows the results for SGL-25BC base and aluminum
sidewall. All the three different behaviors, droplet jumping, corner
filling and non-filling of the channel corners were observed for different open angles used. Droplet jumping behavior was observed
at 45◦ , 50◦ and 60◦ open angles, which are not near the transition
angle for this material pair. However, using the proposed criterion,
the droplet jumping was predicted accurately. From the experiments, it was seen that droplet filled the channel corners for open
angles above 60◦ and did not fill the channel corners for corner
angles below 60◦ . The experimental results were in accordance
with Concus–Finn condition.
Similar results were also obtained for a SGL-24BC base and rough
polycarbonate sidewall. Experimental results are shown in Fig. 5(f).
Here also all the three different behaviors, droplet jumping, corner filling and non-filling of the channel corners were observed
for different channel open angles used. Droplets showed jumping
behavior at 50◦ channel angle. Using the proposed criterion, the
droplet jumping behavior was predicted accurately. Droplets also
showed corner filling behavior at angles above 60◦ and no corner filling at angles below 50◦ , which are in accordance with the
Concus–Finn condition.
Comparing the R2 /RC2 data results for all the material pairs used
it was seen that for more than 96% of the cases the theoretical
prediction for droplet jumping behavior matched the experimental data. To understand why the droplet jumps to the sidewall
an energy balance on the droplet while sitting on the base wall
and while hanging from the sidewall was performed. This is similar to the droplet spreading analysis performed on a surface to
understand the wetting behavior of a droplet [43–52]. Detailed
explanation of the energy balance and the results are presented
in section 4.
It is noted from the proposed model based on geometric analysis that the droplet jumping behavior depends on the channel
open angle, material property and droplet emergence location in
the channel. Among all these parameters, droplet emergence location plays a major role in the droplet behavior near the channel
corner. When the droplet emerges very close to the channel corner, the droplet does not have enough room to jump and hang from
the sidewall. Instead, it would touch the sidewall and form a bridge
between the sidewall and the base as shown in Fig. 6.
Fig. 6 shows the droplet jumping image sequence for an SGL25BC base and an aluminum sidewall at 45◦ corner angle where
the droplet emergence is very close to the channel corner. Fig. 6(a)
shows the emergence of a droplet from the GDL base very close to
the channel corner. Fig. 6(b) shows the image of the droplet when it
contacts the sidewall right before it jumps. Once the contact is made
with the sidewall, the droplet quickly moves toward the sidewall as
shown in Fig. 6(c) and then jumps as shown in Fig. 6(d). However,
due to the smaller clearance between the base and the sidewall
near the channel corner, the droplet bridges between the two walls
as shown in Fig. 6(e). When a droplet emerges very near to the
channel corner (less than 0.1 mm away from the corner) or under
the land area, it leads to corner filling behavior [15]. Corner filling
is an important issue in a PEMFC gas channels and determines the
two-phase flow patterns and the oxygen transport resistance at the
GDL-channel interface. For this problem, the PEMFC gas channel
design needs to be modified such that the water emerging from
the GDL surface into the channel is removed efficiently.
The authors’ group has been working on experiments and
numerical simulations to understand the effect of water accumulation on the droplet behavior [15,16], area coverage ratio [55] and
the resulting mass transport losses [53,54] in a working PEM fuel
cell. The current work is done from a more fundamental perspective where the air flow is not taken into account. However, further
work is underway to address the effect of air flow and material
properties on droplet dynamics.
4. Explanation of droplet jumping through energy
considerations
At any given time, a droplet would prefer to remain in a lower
energy state. Therefore, for the droplet to jump to the sidewall, the
energy of the droplet while it sits on the base should be higher than
when it hangs from the sidewall, so that it favors to detach from the
base and move toward the sidewall. To understand how the surface
energies affect the droplet jumping behavior, the Gibbs free energy
for the droplet under the two states (a) just before the jump and (b)
the final hanging position were calculated for the conditions that
showed droplet jumping behavior. The total surface energy of the
droplet is expressed as follows.
G0 = LV ALV − ASL cos r
cos r =
(SV − SL )eff
LV
(11)
(12)
11.16
4.03
24.02
3.66
59.96
21.84
13.59
23.12
30.65
2.84
139.62
50.06
18.16
28.74
39.25
5.70
127.24
54.20
42.91
55.89
93.92
12.20
326.82
126.10
4.05
4.17
4.282
4.09
4.91
4.01
0.517
0.719
0.845
0.164
2.506
1.385
0.718
0.805
0.877
0.312
1.512
1.192
0.421
0.516
0.562
0.414
1.071
0.813
0.556
0.819
1.097
0.218
3.836
2.083
Droplet
Radius R2
(mm)
Droplet
radius RC1
(mm)
271
where LV represents the surface tension of the liquid–vapor interface, ALV and ASL represent the surface area of the liquid–vapor
interface and the solid liquid interface of the droplet respectively
[56,57]. For a composite surface the r = rC , and for a wetted contact surface r = rW . As the experiments were performed at room
temperature, the corresponding surface tension LV value used was
72.8 × 10−3 N/m. Using Eqs. (11) and (12), the surface energy for
conditions 1 and 2 were found. The effect of deformation due to
gravity was neglected due to the small droplet size as shown in
Fig. 3 [13]. For the droplet to jump from the base to the sidewall,
there is some amount of work done against the gravity. An energy
balance for the two states is given by the following equation.
G0,Droplet on base = G0,Hanging droplet + EGravity + Losses
(13)
where G0 , Droplet on base = Gibbs free energy when the droplet is
sitting on the base.
GO , Hanging Droplet = Gibbs free energy when the droplet is hanging
from the sidewall.
EGravity = Energy required by the droplet to move from the base
to the sidewall against gravity (potential energy).
Losses = Losses mainly due to viscous dissipation.
For calculating EGravity , the center-of-mass of the droplet while
sitting on the base and when hanging from the sidewall are determined. The vertical distance between the two center-of-masses is
then calculated and used as the height the droplet moved against
the gravity. Equating all the different energy terms, the energy loss
due to viscous dissipation is calculated. The different energy values
calculated for different test conditions where the droplet jumped
to the sidewall are shown in Table 3. The radii for different material
pairs when the droplet is sitting on the base surface and when it
is hanging from the sidewall are given by RC1 and R2 , respectively.
Using these radii, the surface area of the liquid–vapor interface of
the droplet is found. From the surface area values, the Gibbs free
energy of the droplet while sitting on the base and while hanging
from the sidewall is determined. It is seen from the Table 3 that the
energy while sitting on the base was almost 1.5 times greater than
when the droplet was hanging from the sidewall. This shows that
the droplet prefers the lower energy state, which makes the droplet
to jump to the sidewall. The loss terms are positive and quite small;
indicating that the energy approach provides a good explanation
for the droplet jumping condition. The maximum energy losses are
around 15% of the Gibbs free energy when the droplet is sitting on
the base. These losses are mainly due to the oscillations the droplet
undergoes when it jumps to the sidewall before it stabilizes and
due to the viscous dissipations of the droplet.
1.081
1.04
1.133
0.91
2.16
1.84
40
50
60
30
50
60
SGL-polycarbonate
SGL-polycarbonate
SGL-polycarbonate
PTFE-polycarbonate
SGL-aluminum
SGL-aluminum
1
2
3
4
5
6
Pore
distance
(mm)
Material
Open
angle
5. Conclusions
SI. No.
Table 3
Different energies associated with the droplet while jumping to the sidewall.
Surface area of
droplet on
base(mm2 )
Surface area of
hanging
droplet (mm2 )
Droplet Jump
height (mm)
G0 , Droplet on
(J) × 10−9
base
G0 , Hanging
(J) × 10−9
droplet
EGravity
(J) × 10−9
Losses
(J) × 10−9
P. Gopalan, S.G. Kandlikar / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 262–274
Water removal from the gas channels of PEMFCs has been a
major issue of concern from the water management perspective
in automotive fuel cell application. Trapezoidal gas channels are
generally employed because of their desirable two-phase pressure drop and manufacturability features. An earlier study by the
authors’ group showed that the droplet interaction with the sidewall of the gas channel would lead to jumping of the droplet to
the sidewall, and fill, or not-fill conditions at the channel corners.
In this work, a theoretical model was presented to capture the
droplet-wall interactions in terms of the droplet jumping, corner
filling, and non-filling behaviors. It was seen from the model that
a droplet emerging from the horizontal base jumps to the sidewall
of the gas channel under certain conditions of the trapezoid corner
angle, surface energies of the channel wall and the base material,
and the droplet emergence location. This work was complemented
with an experimental study for understanding the droplet dynamics in terms of the location of the droplet emergence, the corner
272
P. Gopalan, S.G. Kandlikar / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 262–274
angle, and the surface energies of the walls. The proposed model is
able to predict the droplet interaction behaviors with the sidewall
of the channel accurately.
Rearranging Eq. (1), the maximum radius of the droplet just
before it snaps and jumps to the sidewall is obtained.
RC1 =
Acknowledgments
This work was conducted in the Thermal Analysis, Microfluidics, and Fuel Cell Laboratory in the Department of Mechanical
Engineering at the Rochester Institute of Technology and was
supported by the US Department of Energy under contract No.
DE-EE0000470.
a tan 2˛
sin ˇ + sin ı + cos ı tan 2˛
(30)
Substituting ˇ and ı in terms of and 2˛, the critical radius
RC1 can be obtained.
RC1 =
a tan 2˛
sin − /2 + sin (/2 − 2˛) + cos (/2 − 2˛) tan 2˛
(31)
Appendix I.
The following equations are derived from the geometrical relationships as shown in Figs. 2 and 4.
ˇ=−
2
(14)
cos ˇ = cos −
ı=
2
= sin (15)
− 2˛
2
cos ı = cos
(16)
− 2˛
= sin 2˛
Solving Eq. (31), RC1 is obtained as below
RC1 =
a sin 2˛
1 − (cos cos 2˛)
(2) To find the distance of the point of contact of the droplet on the
sidewall (x) from the channel corner before it jumps
From Fig. 2,
2
x = [(b + d) + (a − e)2 ]
1/2
(32)
(17)
Substituting values of b, d and e in Eq. (32)
d = RC1 sin ı
(18)
x = [(RC1 sin ˇ + RC1 sin ı) + (a − RC1 cos ı) ]
e = RC1 cos ı
(19)
b = RC1 sin ˇ
(20)
2
h = RC1 − b = RC1 (1 − sin ˇ) = RC1
h = RC1 (1 + cos )
1 − sin −
2
(21)
(22)
− 2˛
2
(23)
k=
−
2
(24)
z = RC2 sin k = RC2 cos f = z tan 2˛ = RC2 cos tan 2˛
z
g
(25)
Substituting ˇ and ı in terms of and 2˛,
2
x = [RC2 (cos 2˛ − cos ) + (a − RC1 sin 2˛)2 ]
RC cos z
= 2
g=
cos 2˛
sin (28)
m = R2 − z = R2 − R2 cos (29)
(3) To find the volume (V1 ) of the droplet sitting on the GDL
Volume of the droplet when it is sitting on the base surface
right before it touches the sidewall as shown in Fig. 2 is given
as
Volume of the droplet for case 1 as shown in Fig. 4 is given as
V1 = Vsphere − Vchord
(34)
4RC3
Vsphere =
Vchord =
1
(35)
3
V1 =
V1 =
(1) To find the maximum radius RC1 right before it jumped
The open angle 2˛ is related to the RC1 and is given by Eq. (1)
h2 (3RC1 − h)
(36)
3
Where h is the height of the spherical cap as shown in Fig. 2
Substituting Eqs. (35) and (36) in Eq. (34)
Appendix II.
4RC3
1
3
−
h2 (3RC1 − h)
(37)
3
4RC3 − 3RC1 h2 + h3 )
1
(38)
3
Substituting h in Eq. (38)
2
tan 2˛ =
a − RC1 cos ı
1/2
1
(26)
(27)
RC1 sin ˇ + RC1 sin ı
(33)
=
sin =
2 1/2
2
V1 =
4RC3 − 3RC1 (RC1 + RC1 cos ) + (RC1 + RC1 cos )
3
1
3
(39)
P. Gopalan, S.G. Kandlikar / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 262–274
273
2
V1 =
4RC3 − 3RC1 (RC2 + RC2 cos2 + 2RC2 cos ) + (RC3 + RC3 cos3 + 3RC3 cos + 3RC3 cos2 )
1
1
1
1
1
1
1
1
3
(40)
Rearranging Eq. (40)
V1 =
V1 =
(4RC3 − 3RC3 − 3RC3 cos2 − 6RC3 cos + RC3 + RC3 cos3 + 3RC3 cos + 3RC3 cos2 )
1
1
1
1
1
1
1
3
1
1
1
3
2
m (3R2 − m)
3
(42)
Substituting m in Eq. (42)V2 = 3 R23 (2 − 3 cos + cos3 )
(5) To find the maximum radius RC2 while hanging from the sidewall before bridging
Fig. 4 shows the triangle made by point A, B and C from which
the relationship between the radius RC2 , midpoint x around
which the droplet hangs and the open angle 2˛ is found.
sin 2˛ =
RC2 − g
(43)
x−f
Substituting Eqs. (26) and (28) in Eq. (43)
sin 2˛ =
RC2 − (RC2 cos / cos 2˛)
x − RC2 cos tan 2˛
Eq. (7) is rearranged to get RC2 in terms of open angle and
sidewall contact angle
RC2 =
x sin 2˛
(1 − cos cos 2˛)
(44)
Substituting x in Eq. (44)
2
RC2 =
(41)
(2RC3 − 3RC3 cos + RC3 cos 3 )
(4) To find the volume (V2 ) of the droplet hanging from the sidewall
Volume of the droplet when it jumps to the wall as shown in
Fig. 4 is given as
V2 =
1
[RC2 (cos 2˛ − cos ) + (a − RC1 sin 2˛)2 ]
1
1/2
sin 2˛
(1 − cos cos 2˛)
Therefore, Eq. (8) gives RC2 in terms of the open angle, droplet
emergence location, and base and sidewall contact angles.
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